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Difficulty: Very hardRatios, Rates, and Proportions

Three industrial machines, XX, YY, and ZZ, continuously produce liquid chemical mixtures composed solely of Compounds AA and BB.

- Machine XX produces the mixture at a constant rate of 100100 liters per hour, with Compound AA and Compound BB in a ratio of 3:23:2 by volume.
- Machine YY produces the mixture at a constant rate of 150150 liters per hour, with Compound AA and Compound BB in a ratio of 1:41:4 by volume.
- Machine ZZ produces the mixture at a constant rate of RR liters per hour, with Compound AA and Compound BB in a ratio of 7:37:3 by volume.

If all three machines operate simultaneously for 44 hours, the resulting combined mixture contains equal total volumes of Compound AA and Compound BB.

Which of the following statements must be true? Select all that apply.

  1. The operating rate RR of Machine ZZ is 175175 liters per hour.Answer
  2. Machine ZZ accounts for more than 40%40\% of the total volume of the combined mixture produced in 44 hours.Answer
  3. In the 4-hour output, the volume of Compound BB produced by Machine YY is greater than the total volume of Compound BB produced by Machines XX and ZZ combined.Answer
  4. D
    The ratio of the volume of Compound AA produced by Machine XX to the volume of Compound AA produced by Machine ZZ is 2:32:3.
  5. E
    If Machine YY were turned off and only Machines XX and ZZ operated for 44 hours, the combined output would have Compound AA and Compound BB in a ratio of 7:37:3.

Answer

The statements asserting that Machine Z's rate is 175 liters per hour, that Machine Z accounts for over 40% of the total output volume, and that Machine Y produces more Compound B than Machines X and Z combined are all true.
The correct options are those stating that Machine Z operates at 175 liters per hour, that Machine Z accounts for over 40% of the 4-hour output, and that Machine Y produces more Compound B than Machines X and Z combined. Machine X produces 240 liters of Compound A and 160 liters of Compound B in 4 hours. Machine Y produces 120 liters of Compound A and 480 liters of Compound B in 4 hours. Machine Z produces 2.8R liters of Compound A and 1.2R liters of Compound B in 4 hours. Equating total Compound A (360 + 2.8R) to total Compound B (640 + 1.2R) gives 1.6R = 280, so R = 175 liters per hour. Machine Z thus contributes 700 liters out of 1700 total liters, which is approximately 41.18% (greater than 40%). Machine Y produces 480 liters of Compound B, which exceeds the 370 liters of Compound B produced by Machines X and Z combined (160 + 210 liters).

Step-by-Step Solution

1
Calculate the volumes of Compound A and Compound B produced by Machines X and Y in 4 hours.
Machine X produces 400400 L total (240240 L of A, 160160 L of B). Machine Y produces 600600 L total (120120 L of A, 480480 L of B). Combined from X and Y: 360360 L of A and 640640 L of B.
Decompose each machine's total 4-hour output into individual compound amounts using part-to-whole fraction multipliers.
2
Formulate Machine Z's 4-hour output expressions and solve for rate R.
Machine Z produces 4R4R L total (2.8R2.8R L of A, 1.2R1.2R L of B). Setting total A equal to total B: 360+2.8R=640+1.2R  ⟹  1.6R=280  ⟹  R=175360 + 2.8R = 640 + 1.2R \implies 1.6R = 280 \implies R = 175 L/hr.
The condition of equal total volumes of Compound A and Compound B forms a single linear equation in terms of R.
3
Determine Machine Z's proportion of the total volume produced by all three machines.
Machine Z volume = 4×175=7004 \times 175 = 700 L. Total volume = 400+600+700=1700400 + 600 + 700 = 1700 L. Percentage = 7001700×100%≈41.18%>40%\frac{700}{1700} \times 100\% \approx 41.18\% > 40\%.
Evaluate whether Machine Z contributes more than 40% of the complete 1700-liter mixture.
4
Compare Compound B output from Machine Y against the combined Compound B output from X and Z.
Machine Y produces 480480 L of B. Machines X and Z produce 160+(1.2×175)=370160 + (1.2 \times 175) = 370 L of B. 480>370480 > 370 is true.
Direct numerical comparison verifies the inequality statement.
5
Evaluate the remaining ratio claims.
Compound A ratio X to Z is 240:490=24:49≠2:3240 : 490 = 24 : 49 \neq 2 : 3. Combined X and Z ratio of A to B is 730:370=73:37≠7:3730 : 370 = 73 : 37 \neq 7 : 3. Both claims are false.
Check remaining distractors for mathematical accuracy.

Key Concept

Multi-rate and multi-component ratio mixture balancing
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Topics

Ratios, Rates, and Proportions

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